It is known that \(t \mapsto E_{\alpha ,1}(-\lambda t^{\alpha })\)  and \(t \mapsto t^{\beta -1} E_{\alpha ,\beta }(-\lambda t^{\alpha })\)  are completely monotone on \([0,\infty )\) , see, e.g., [1]}. Since they are not identically zero, they are positive on \([0,\infty )\) . By Lemma REF , these functions are log-convex on \([0,\infty )\) .
